Are there examples of conjectures that turned out to be false despite of strong heuristics? Several open conjectures are widely believed to be true due to strong heuristical evidence. Examples include Goldbach's conjecture , Collatz's conjecture and the Riemann hypothesis.

Are there examples of conjectures that turned out to be false despite of a strong heuristical evdidence to be true ?

I know that there are conjectures with very large smallest counterexamples , but I wonder whether there were additional reasons (except that a counterexample wasn't found for a long time) to believe the conjecture. 
 A: There is a conjecture that the number of primes between $x$ and $x+y$, inclusive, is never more than the number of primes between $2$ and $2+y$, inclusive. It seems reasonable, since anyone can see that the primes thin out as you go up, and it has never been disproved, but forty years ago it was proved to be in conflict with another unproved conjecture that has even better heuristic evidence, so it's now believed to be false. 
A: They were rarely called conjectures, but when General Relativity was proposed by Einstein it was expected that solutions would be "physically reasonable".  Much later Godel discovered an exact solution, which allowed time travel into the past among other things.  Later it was thought that singularities were a consequence of the high degree of symmetry of solutions, such as the Schwarzschild black hole solution, but later Hawking and others proved that singularities arise under fairly general conditions.  "Cosmic Censorship", the idea that singularities are always hidden behind event horizons and never visible to us (although the big bang is essentially a visible singularity) is still a conjecture of classical General Relativity.  Hawking showed that when quantum effects are included, black holes emit radiation (contrary to the belief that nothing can escape) and a black hole can "evaporate", but may create a visible singularity as it dissapears.
